Question

Two right circular cylinders have the same volume. The radius of the second cylinder is $10\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders?

The second height is $10\%$ less than the first.

The first height is $10\%$ more than the second.

The second height is $21\%$ less than the first.

The first height is $21\%$ more than the second.

The second height is $80\%$ of the first.

Solution:

Let $r,h$ and $R,H$ be the radii and heights of the first and second cylinders. The volumes are equal, so $\pi r^2 h = \pi R^2 H,$ and $R = 1.1r.$

Then $\pi r^2 h = \pi(1.1r)^2 H = \pi(1.21 r^2) H.$ Dividing by $\pi r^2$ yields $h = 1.21H,$ so the first height is $21\%$ more than the second.

Thus, the correct answer is D.

2015 AMC 12A Problem 7

Solution:

Problem 7 in Other Years